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"... There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the con ..."

There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, Gödel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit ” (in German) or “finitary ” or “finitistic ” primarily to refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [Gödel, 1938a] and the lecture notes for a lecture at Yale [Gödel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of first-order number theory, P A; but starting in the Dialectica paper

...for pointing this out to me so that I avoided error (at least about this). As to the precise relationship between Gödel and Turing on computability and the issues involved, the reader should consult [=-=Sieg, 2006-=-]. 20of recursion up to α.) But I don’t see why an arbitrary finitary consistency proof for P A should translate into Gentzen’s. Nevertheless, the argument that recursion on ordinals < ɛ0 is not fini...

"... Abstract: Many philosophers contend that Turing’s work provides a conceptual analysis of numerical computability. In (Rescorla, 2007), I dissented. I argued that the problem of deviant notations stymies existing attempts at conceptual analysis. Copeland and Proudfoot respond to my critique. I argue ..."

Abstract: Many philosophers contend that Turing’s work provides a conceptual analysis of numerical computability. In (Rescorla, 2007), I dissented. I argued that the problem of deviant notations stymies existing attempts at conceptual analysis. Copeland and Proudfoot respond to my critique. I argue that their putative solution does not succeed. We are still awaiting a genuine conceptual analysis.

...ons of Real Variables contains one of the earliest examples of “recursion formula” in the English language. More to the point of this thesis, Kurt Gödel writes “rekursiv” in German in 1931 (cited in =-=Sieg 2006-=-) to refer to the class of functions that were to become so central to mathematical logic. This central role revolves around the original mathematical interpretation of 9 UNIVERSITAT ROVIRA I VIRGILIs...

"... Infons are statements viewed as containers of information (rather then representations of truth values). The logic of infons turns out to be a conservative extension of logic known as constructive or intuitionistic. Distributed Knowledge Authorization Language uses additional unary connectives “p sa ..."

Infons are statements viewed as containers of information (rather then representations of truth values). The logic of infons turns out to be a conservative extension of logic known as constructive or intuitionistic. Distributed Knowledge Authorization Language uses additional unary connectives “p said ” and “p implied ” where p ranges over principals. Here we investigate infon logic and a narrow but useful primal fragment of it. In both cases, we develop model theory and analyze the derivability problem: Does the given query follow from the given hypotheses? Our more involved technical results are on primal infon logic. We construct an algorithm for the multiple derivability problem: Which of the given queries follow from the given hypotheses? Given a bound on the quotation depth of the hypotheses, the algorithm runs in linear time. We quickly discuss the significance of this result for access control.

"... Abstract. We put the title problem and Church’s thesis into a proper perspective, and we address some common misconceptions about Turing’s analysis of computation. In addition, we comment on two approaches to the title problem, one well known among philosophers and another well known among logicians ..."

Abstract. We put the title problem and Church’s thesis into a proper perspective, and we address some common misconceptions about Turing’s analysis of computation. In addition, we comment on two approaches to the title problem, one well known among philosophers and another well known among logicians.

...hard to isolate first principles that, in Turing’s opinion, are satisfied by all symbolic sequential computations. Wilfried Sieg adopted Gandy’s approach and reworked Gandy’s axioms to an extent; see =-=[28]-=- and references there. For our purposes here, there is no essential difference between Gandy’s original axioms and Sieg’s versions of the axioms. Critical remarks. In a 2002 article [25], Oron Shagrir...

"... The real question at issue is “What are the possible processes which can be carried out in computing a number?” Turing Give me a fulcrum, and I shall move the world. Archimedes Abstract. Alan Turing pioneered semantics-to-syntax analysis of algo-rithms. It is a kind of analysis where you start with ..."

The real question at issue is “What are the possible processes which can be carried out in computing a number?” Turing Give me a fulcrum, and I shall move the world. Archimedes Abstract. Alan Turing pioneered semantics-to-syntax analysis of algo-rithms. It is a kind of analysis where you start with a large semantically defined species of algorithms, and you finish up with a syntactic artifact, typically a computation model, that characterizes the species. The task of analyzing a large species of algorithms seems daunting if not impossi-ble. As in quicksand, one needs a rescue point, a fulcrum. In computation analysis, a fulcrum is a particular viewpoint on computation that clari-fies and simplifies things to the point that analysis become possible. We review from that point of view Turing’s analysis of human-executable computation, Kolmogorov’s analysis of sequential bit-level computation, Gandy’s analysis of a species of machine computation, and our own anal-ysis of sequential computation.

...e step, and the value depends on the whole input. Our additional critical remarks of Gandy’s analysis are found in [16, §4]. (Wilfried Sieg adopted Gandy’s approach and simplified Gandy’s axioms, see =-=[24]-=- and references there, but — as far as our critique is concerned — the improvements do not make much difference.) Q: If Turing thought of synchronous parallelism, he could have claimed that, without l...